
Physica D: Nonlinear Phenomena [SJR: 1.049] [HI: 102] [3 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 01672789 Published by Elsevier [3038 journals] 
 Causal hydrodynamics from kinetic theory by doublet scheme in
renormalizationgroup method Authors: Kyosuke Tsumura; Yuta Kikuchi; Teiji Kunihiro
Pages: 1  27
Abstract: Publication date: 1 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 336
Author(s): Kyosuke Tsumura, Yuta Kikuchi, Teiji Kunihiro
We develop a general framework in the renormalizationgroup (RG) method for extracting a mesoscopic dynamics from an evolution equation by incorporating some excited (fast) modes as additional components to the invariant manifold spanned by zero modes. We call this framework the doublet scheme. The validity of the doublet scheme is first tested and demonstrated by taking the Lorenz model as a simple threedimensional dynamical system; it is shown that the twodimensional reduced dynamics on the attractive manifold composed of the wouldbe zero and a fast modes are successfully obtained in a natural way. We then apply the doublet scheme to construct causal hydrodynamics as a mesoscopic dynamics of kinetic theory, i.e., the Boltzmann equation, in a systematic manner with no adhoc assumption. It is found that our equation has the same form as Grad’s thirteenmoment causal hydrodynamic equation, but the microscopic formulae of the transport coefficients and relaxation times are different. In fact, in contrast to the Grad equation, our equation leads to the same expressions for the transport coefficients as given by the Chapman–Enskog expansion method and suggests novel formulae of the relaxation times expressed in terms of relaxation functions which allow a natural physical interpretation of the relaxation times. Furthermore, our theory nicely gives the explicit forms of the distribution function and the thirteen hydrodynamic variables in terms of the linearized collision operator, which in turn clearly suggest the proper ansatz forms of them to be adopted in the method of moments.
PubDate: 20161016T12:47:09Z
DOI: 10.1016/j.physd.2016.06.012
Issue No: Vol. 336 (2016)
 Authors: Kyosuke Tsumura; Yuta Kikuchi; Teiji Kunihiro
 Towards the modeling of nanoindentation of virus shells: Do substrate
adhesion and geometry matter? Authors: Arthur Bousquet; Bogdan Dragnea; Manel Tayachi; Roger Temam
Pages: 28  38
Abstract: Publication date: 1 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 336
Author(s): Arthur Bousquet, Bogdan Dragnea, Manel Tayachi, Roger Temam
Soft nanoparticles adsorbing at surfaces undergo deformation and buildup of elastic strain as a consequence of interfacial adhesion of similar magnitude with constitutive interactions. An example is the adsorption of virus particles at surfaces, a phenomenon of central importance for experiments in virus nanoindentation and for understanding of virus entry. The influence of adhesion forces and substrate corrugation on the mechanical response to indentation has not been studied. This is somewhat surprising considering that many singlestranded RNA icosahedral viruses are organized by soft intermolecular interactions while relatively strong adhesion forces are required for virus immobilization for nanoindentation. This article presents numerical simulations via finite elements discretization investigating the deformation of a thick shell in the context of slow evolution linear elasticity and in presence of adhesion interactions with the substrate. We study the influence of the adhesion forces in the deformation of the virus model under axial compression on a flat substrate by comparing the force–displacement curves for a shell having elastic constants relevant to virus capsids with and without adhesion forces derived from the LennardJones potential. Finally, we study the influence of the geometry of the substrate in twodimensions by comparing deformation of the virus model adsorbed at the cusp between two cylinders with that on a flat surface.
PubDate: 20161016T12:47:09Z
DOI: 10.1016/j.physd.2016.06.013
Issue No: Vol. 336 (2016)
 Authors: Arthur Bousquet; Bogdan Dragnea; Manel Tayachi; Roger Temam
 Fluctuations induced transition of localization of granular objects caused
by degrees of crowding Authors: Soutaro Oda; Yoshitsugu Kubo; ChwenYang Shew; Kenichi Yoshikawa
Pages: 39  46
Abstract: Publication date: 1 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 336
Author(s): Soutaro Oda, Yoshitsugu Kubo, ChwenYang Shew, Kenichi Yoshikawa
Fluctuations are ubiquitous in both microscopic and macroscopic systems, and an investigation of confined particles under fluctuations is relevant to how living cells on the earth maintain their lives. Inspired by biological cells, we conduct the experiment through a very simple fluctuating system containing one or several large spherical granular particles and multiple smaller ones confined on a cylindrical dish under vertical vibration. We find a universal behavior that large particles preferentially locate in cavity interior due to the fact that large particles are depleted from the cavity wall by small spheres under vertical vibration in the actual experiment. This universal behavior can be understood from the standpoint of entropy.
PubDate: 20161016T12:47:09Z
DOI: 10.1016/j.physd.2016.06.014
Issue No: Vol. 336 (2016)
 Authors: Soutaro Oda; Yoshitsugu Kubo; ChwenYang Shew; Kenichi Yoshikawa
 Coupled oscillators on evolving networks
 Authors: R.K. Singh; Trilochan Bagarti
Pages: 47  52
Abstract: Publication date: 1 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 336
Author(s): R.K. Singh, Trilochan Bagarti
In this work we study coupled oscillators on evolving networks. We find that the steady state behavior of the system is governed by the relative values of the spread in natural frequencies and the global coupling strength. For coupling strong in comparison to the spread in frequencies, the system of oscillators synchronize and when coupling strength and spread in frequencies are large, a phenomenon similar to amplitude death is observed. The network evolution provides a mechanism to build interoscillator connections and once a dynamic equilibrium is achieved, oscillators evolve according to their local interactions. We also find that the steady state properties change by the presence of additional time scales. We demonstrate these results based on numerical calculations studying dynamical evolution of limitcycle and van der Pol oscillators.
PubDate: 20161016T12:47:09Z
DOI: 10.1016/j.physd.2016.06.015
Issue No: Vol. 336 (2016)
 Authors: R.K. Singh; Trilochan Bagarti
 Emergence of chaos in a spatially confined reactive system
 Authors: Valérie Voorsluijs; Yannick De Decker
Pages: 1  9
Abstract: Publication date: 15 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 335
Author(s): Valérie Voorsluijs, Yannick De Decker
In spatially restricted media, interactions between particles and local fluctuations of density can lead to important deviations of the dynamics from the unconfined, deterministic picture. In this context, we investigated how molecular crowding can affect the emergence of chaos in small reactive systems. We developed to this end an amended version of the Willamowski–Rössler model, where we account for the impenetrability of the reactive species. We analyzed the deterministic kinetics of this model and studied it with spatiallyextended stochastic simulations in which the mobility of particles is included explicitly. We show that homogeneous fluctuations can lead to a destruction of chaos through a fluctuationinduced collision between chaotic trajectories and absorbing states. However, an interplay between the size of the system and the mobility of particles can counterbalance this effect so that chaos can indeed be found when particles diffuse slowly. This unexpected effect can be traced back to the emergence of spatial correlations which strongly affect the dynamics. The mobility of particles effectively acts as a new bifurcation parameter, enabling the system to switch from stationary states to absorbing states, oscillations or chaos.
PubDate: 20161002T03:34:48Z
DOI: 10.1016/j.physd.2016.05.005
Issue No: Vol. 335 (2016)
 Authors: Valérie Voorsluijs; Yannick De Decker
 Breather solutions for inhomogeneous FPU models using Birkhoff normal
forms Authors: Francisco MartínezFarías; Panayotis Panayotaros
Pages: 10  25
Abstract: Publication date: 15 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 335
Author(s): Francisco MartínezFarías, Panayotis Panayotaros
We present results on spatially localized oscillations in some inhomogeneous nonlinear lattices of Fermi–Pasta–Ulam (FPU) type derived from phenomenological nonlinear elastic network models proposed to study localized protein vibrations. The main feature of the FPU lattices we consider is that the number of interacting neighbors varies from site to site, and we see numerically that this spatial inhomogeneity leads to spatially localized normal modes in the linearized problem. This property is seen in 1D models, and in a 3D model with a geometry obtained from protein data. The spectral analysis of these examples suggests some nonresonance assumptions that we use to show the existence of invariant subspaces of spatially localized solutions in quartic Birkhoff normal forms of the FPU systems. The invariant subspaces have an additional symmetry and this fact allows us to compute periodic orbits of the quartic normal form in a relatively simple way.
PubDate: 20161002T03:34:48Z
DOI: 10.1016/j.physd.2016.06.004
Issue No: Vol. 335 (2016)
 Authors: Francisco MartínezFarías; Panayotis Panayotaros
 Kinetic theory of cluster dynamics
 Authors: Robert I.A. Patterson; Sergio Simonella; Wolfgang Wagner
Pages: 26  32
Abstract: Publication date: 15 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 335
Author(s): Robert I.A. Patterson, Sergio Simonella, Wolfgang Wagner
In a Newtonian system with localized interactions the whole set of particles is naturally decomposed into dynamical clusters, defined as finite groups of particles having an influence on each other’s trajectory during a given interval of time. For an ideal gas with shortrange intermolecular force, we provide a description of the cluster size distribution in terms of the reduced Boltzmann density. In the simplified context of Maxwell molecules, we show that a macroscopic fraction of the gas forms a giant component in finite kinetic time. The critical index of this phase transition is in agreement with previous numerical results on the elastic billiard.
PubDate: 20161002T03:34:48Z
DOI: 10.1016/j.physd.2016.06.007
Issue No: Vol. 335 (2016)
 Authors: Robert I.A. Patterson; Sergio Simonella; Wolfgang Wagner
 Frequency locking near the gluing bifurcation: Spintorque oscillator
under periodic modulation of current Authors: Michael A. Zaks; Arkady Pikovsky
Pages: 33  44
Abstract: Publication date: 15 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 335
Author(s): Michael A. Zaks, Arkady Pikovsky
We consider entrainment by periodic force of limit cycles which are close to the homoclinic bifurcation. Taking as a physical example the nanoscale spintorque oscillator in the LC circuit, we develop the general description of the situation in which the frequency of the stable periodic orbit in the autonomous system is highly sensitive to minor variations of the parameter, and derive explicit expressions for the strongly deformed borders of the resonance regions (Arnold tongues) in the parameter space of the problem. It turns out that proximity to homoclinic bifurcations hinders synchronization of spintorque oscillators.
PubDate: 20161002T03:34:48Z
DOI: 10.1016/j.physd.2016.06.008
Issue No: Vol. 335 (2016)
 Authors: Michael A. Zaks; Arkady Pikovsky
 Chaotic subdynamics in coupled logistic maps
 Authors: Marek Lampart; Piotr Oprocha
Pages: 45  53
Abstract: Publication date: 15 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 335
Author(s): Marek Lampart, Piotr Oprocha
We study the dynamics of Laplaciantype coupling induced by logistic family f μ ( x ) = μ x ( 1 − x ) , where μ ∈ [ 0 , 4 ] , on a periodic lattice, that is the dynamics of maps of the form F ( x , y ) = ( ( 1 − ε ) f μ ( x ) + ε f μ ( y ) , ( 1 − ε ) f μ ( y ) + ε f μ ( x ) ) where ε > 0 determines strength of coupling. Our main objective is to analyze the structure of attractors in such systems and especially detect invariant regions with nontrivial dynamics outside the diagonal. In analytical way, we detect some regions of parameters for which a horseshoe is present; and using simulations global attractors and invariant sets are depicted.
PubDate: 20161002T03:34:48Z
DOI: 10.1016/j.physd.2016.06.010
Issue No: Vol. 335 (2016)
 Authors: Marek Lampart; Piotr Oprocha
 A hierarchy of Poisson brackets in nonequilibrium thermodynamics
 Authors: Michal Pavelka; Václav Klika; Oğul Esen; Miroslav Grmela
Pages: 54  69
Abstract: Publication date: 15 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 335
Author(s): Michal Pavelka, Václav Klika, Oğul Esen, Miroslav Grmela
Reversible evolution of macroscopic and mesoscopic systems can be conveniently constructed from two ingredients: an energy functional and a Poisson bracket. The goal of this paper is to elucidate how the Poisson brackets can be constructed and what additional features we also gain by the construction. In particular, the Poisson brackets governing reversible evolution in oneparticle kinetic theory, kinetic theory of binary mixtures, binary fluid mixtures, classical irreversible thermodynamics and classical hydrodynamics are derived from Liouville equation. Although the construction is quite natural, a few examples where it does not work are included (e.g. the BBGKY hierarchy). Finally, a new infinite grandcanonical hierarchy of Poisson brackets is proposed, which leads to Poisson brackets expressing nonlocal phenomena such as turbulent motion or evolution of polymeric fluids. Eventually, Lie–Poisson structures standing behind some of the brackets are identified.
PubDate: 20161002T03:34:48Z
DOI: 10.1016/j.physd.2016.06.011
Issue No: Vol. 335 (2016)
 Authors: Michal Pavelka; Václav Klika; Oğul Esen; Miroslav Grmela
 Topology in Dynamics, Differential Equations, and Data
 Authors: Sarah Day; Robertus C.A.M. Vandervorst; Thomas Wanner
Pages: 1  3
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Sarah Day, Robertus C.A.M. Vandervorst, Thomas Wanner
This special issue is devoted to showcasing recent uses of topological methods in the study of dynamical behavior and the analysis of both numerical and experimental data. The twelve original research papers span a wide spectrum of results from abstract index theories, over homology and persistencebased data analysis techniques, to computerassisted proof techniques based on topological fixed point arguments.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.08.003
Issue No: Vol. 334 (2016)
 Authors: Sarah Day; Robertus C.A.M. Vandervorst; Thomas Wanner
 Geometric phase in the Hopf bundle and the stability of nonlinear waves
 Authors: Colin J. Grudzien; Thomas J. Bridges; Christopher K.R.T. Jones
Pages: 4  18
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Colin J. Grudzien, Thomas J. Bridges, Christopher K.R.T. Jones
We develop a stability index for the traveling waves of nonlinear reaction–diffusion equations using the geometric phase induced on the Hopf bundle S 2 n − 1 ⊂ C n . This can be viewed as an alternative formulation of the winding number calculation of the Evans function, whose zeros correspond to the eigenvalues of the linearization of reaction–diffusion operators about the wave. The stability of a traveling wave can be determined by the existence of eigenvalues of positive real part for the linear operator. Our method of geometric phase for locating and counting eigenvalues is inspired by the numerical results in Way’s Dynamics in the Hopf bundle, the geometric phase and implications for dynamical systems Way (2009). We provide a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined on C 2 and sketch the proof of the method of geometric phase for C n and its generalization to boundaryvalue problems. Implementing the numerical method, modified from Way (2009), we conclude with open questions inspired from the results.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.04.005
Issue No: Vol. 334 (2016)
 Authors: Colin J. Grudzien; Thomas J. Bridges; Christopher K.R.T. Jones
 The Poincaré–Bendixson Theorem and the nonlinear
Cauchy–Riemann equations Authors: J.B. van den Berg; S. Munaò; R.C.A.M. Vandervorst
Pages: 19  28
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): J.B. van den Berg, S. Munaò, R.C.A.M. Vandervorst
Fiedler and MalletParet (1989) prove a version of the classical Poincaré–Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the nonlinear Cauchy–Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapunov function.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.04.009
Issue No: Vol. 334 (2016)
 Authors: J.B. van den Berg; S. Munaò; R.C.A.M. Vandervorst
 Arnold’s mechanism of diffusion in the spatial circular restricted
threebody problem: A semianalytical argument Authors: Amadeu Delshams; Marian Gidea; Pablo Roldan
Pages: 29  48
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Amadeu Delshams, Marian Gidea, Pablo Roldan
We consider the spatial circular restricted threebody problem, on the motion of an infinitesimal body under the gravity of Sun and Earth. This can be described by a 3degree of freedom Hamiltonian system. We fix an energy level close to that of the collinear libration point L 1 , located between Sun and Earth. Near L 1 there exists a normally hyperbolic invariant manifold, diffeomorphic to a 3sphere. For an orbit confined to this 3sphere, the amplitude of the motion relative to the ecliptic (the plane of the orbits of Sun and Earth) can vary only slightly. We show that we can obtain new orbits whose amplitude of motion relative to the ecliptic changes significantly, by following orbits of the flow restricted to the 3sphere alternatively with homoclinic orbits that turn around the Earth. We provide an abstract theorem for the existence of such ‘diffusing’ orbits, and numerical evidence that the premises of the theorem are satisfied in the threebody problem considered here. We provide an explicit construction of diffusing orbits. The geometric mechanism underlying this construction is reminiscent of the Arnold diffusion problem for Hamiltonian systems. Our argument, however, does not involve transition chains of tori as in the classical example of Arnold. We exploit mostly the ‘outer dynamics’ along homoclinic orbits, and use very little information on the ‘inner dynamics’ restricted to the 3sphere. As a possible application to astrodynamics, diffusing orbits as above can be used to design low cost maneuvers to change the inclination of an orbit of a satellite near L 1 from a nearlyplanar orbit to a tilted orbit with respect to the ecliptic. We explore different energy levels, and estimate the largest orbital inclination that can be achieved through our construction.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.06.005
Issue No: Vol. 334 (2016)
 Authors: Amadeu Delshams; Marian Gidea; Pablo Roldan
 Exploring the topology of dynamical reconstructions
 Authors: Joshua Garland; Elizabeth Bradley; James D. Meiss
Pages: 49  59
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Joshua Garland, Elizabeth Bradley, James D. Meiss
Computing the statespace topology of a dynamical system from scalar data requires accurate reconstruction of those dynamics and construction of an appropriate simplicial complex from the results. The reconstruction process involves a number of free parameters and the computation of homology for a large number of simplices can be expensive. This paper is a study of how to compute the homology efficiently and effectively without a full (diffeomorphic) reconstruction. Using trajectories from the classic Lorenz system, we reconstruct the dynamics using the method of delays, then build a simplicial complex whose vertices are a small subset of the data: the “witness complex”. Surprisingly, we find that the witness complex correctly resolves the homology of the underlying invariant set from noisy samples of that set even if the reconstruction dimension is well below the thresholds for assuring topological conjugacy between the true and reconstructed dynamics that are specified in the embedding theorems. We conjecture that this is because the requirements for reconstructing homology are less stringent: a homeomorphism is sufficient—as opposed to a diffeomorphism, as is necessary for the full dynamics. We provide preliminary evidence that a homeomorphism, in the form of a delaycoordinate reconstruction map, may exist at a lower dimension than that required to achieve an embedding.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.03.006
Issue No: Vol. 334 (2016)
 Authors: Joshua Garland; Elizabeth Bradley; James D. Meiss
 Topological microstructure analysis using persistence landscapes
 Authors: Paweł Dłotko; Thomas Wanner
Pages: 60  81
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Paweł Dłotko, Thomas Wanner
Phase separation mechanisms can produce a variety of complicated and intricate microstructures, which often can be difficult to characterize in a quantitative way. In recent years, a number of novel topological metrics for microstructures have been proposed, which measure essential connectivity information and are based on techniques from algebraic topology. Such metrics are inherently computable using computational homology, provided the microstructures are discretized using a thresholding process. However, while in many cases the thresholding is straightforward, noise and measurement errors can lead to misleading metric values. In such situations, persistence landscapes have been proposed as a natural topology metric. Common to all of these approaches is the enormous data reduction, which passes from complicated patterns to discrete information. It is therefore natural to wonder what type of information is actually retained by the topology. In the present paper, we demonstrate that averaged persistence landscapes can be used to recover central system information in the Cahn–Hilliard theory of phase separation. More precisely, we show that topological information of evolving microstructures alone suffices to accurately detect both concentration information and the actual decomposition stage of a data snapshot. Considering that persistent homology only measures discrete connectivity information, regardless of the size of the topological features, these results indicate that the system parameters in a phase separation process affect the topology considerably more than anticipated. We believe that the methods discussed in this paper could provide a valuable tool for relating experimental data to model simulations.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.04.015
Issue No: Vol. 334 (2016)
 Authors: Paweł Dłotko; Thomas Wanner
 Analysis of Kolmogorov flow and Rayleigh–Bénard convection
using persistent homology Authors: Miroslav Kramár; Rachel Levanger; Jeffrey Tithof; Balachandra Suri; Mu Xu; Mark Paul; Michael F. Schatz; Konstantin Mischaikow
Pages: 82  98
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Miroslav Kramár, Rachel Levanger, Jeffrey Tithof, Balachandra Suri, Mu Xu, Mark Paul, Michael F. Schatz, Konstantin Mischaikow
We use persistent homology to build a quantitative understanding of large complex systems that are driven farfromequilibrium. In particular, we analyze image time series of flow field patterns from numerical simulations of two important problems in fluid dynamics: Kolmogorov flow and Rayleigh–Bénard convection. For each image we compute a persistence diagram to yield a reduced description of the flow field; by applying different metrics to the space of persistence diagrams, we relate characteristic features in persistence diagrams to the geometry of the corresponding flow patterns. We also examine the dynamics of the flow patterns by a second application of persistent homology to the time series of persistence diagrams. We demonstrate that persistent homology provides an effective method both for quotienting out symmetries in families of solutions and for identifying multiscale recurrent dynamics. Our approach is quite general and it is anticipated to be applicable to a broad range of open problems exhibiting complex spatiotemporal behavior.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.02.003
Issue No: Vol. 334 (2016)
 Authors: Miroslav Kramár; Rachel Levanger; Jeffrey Tithof; Balachandra Suri; Mu Xu; Mark Paul; Michael F. Schatz; Konstantin Mischaikow
 Principal component analysis of persistent homology rank functions with
case studies of spatial point patterns, sphere packing and colloids Authors: Vanessa Robins; Katharine Turner
Pages: 99  117
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Vanessa Robins, Katharine Turner
Persistent homology, while ostensibly measuring changes in topology, captures multiscale geometrical information. It is a natural tool for the analysis of point patterns. In this paper we explore the statistical power of the persistent homology rank functions. For a point pattern X we construct a filtration of spaces by taking the union of balls of radius a centred on points in X , X a = ∪ x ∈ X B ( x , a ) . The rank function β k ( X ) : { ( a , b ) ∈ R 2 : a ≤ b } → R is then defined by β k ( X ) ( a , b ) = rank ( ι ∗ : H k ( X a ) → H k ( X b ) ) where ι ∗ is the induced map on homology from the inclusion map on spaces. We consider the rank functions as lying in a Hilbert space and show that under reasonable conditions the rank functions from multiple simulations or experiments will lie in an affine subspace. This enables us to perform functional principal component analysis which we apply to experimental data from colloids at different effective temperatures and to sphere packings with different volume fractions. We also investigate the potential of rank functions in providing a test of complete spatial randomness of 2D point patterns using the distances to an empirically computed mean rank function of binomial point patterns in the unit square.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.03.007
Issue No: Vol. 334 (2016)
 Authors: Vanessa Robins; Katharine Turner
 Continuation of point clouds via persistence diagrams
 Authors: Marcio Gameiro; Yasuaki Hiraoka; Ippei Obayashi
Pages: 118  132
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Marcio Gameiro, Yasuaki Hiraoka, Ippei Obayashi
In this paper, we present a mathematical and algorithmic framework for the continuation of point clouds by persistence diagrams. A key property used in the method is that the persistence map, which assigns a persistence diagram to a point cloud, is differentiable. This allows us to apply the Newton–Raphson continuation method in this setting. Given an original point cloud P , its persistence diagram D , and a target persistence diagram D ′ , we gradually move from D to D ′ , by successively computing intermediate point clouds until we finally find a point cloud P ′ having D ′ as its persistence diagram. Our method can be applied to a wide variety of situations in topological data analysis where it is necessary to solve an inverse problem, from persistence diagrams to point cloud data.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2015.11.011
Issue No: Vol. 334 (2016)
 Authors: Marcio Gameiro; Yasuaki Hiraoka; Ippei Obayashi
 Chaos near a resonant inclinationflip
 Authors: Marcus Fontaine; William Kalies; Vincent Naudot
Pages: 141  157
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Marcus Fontaine, William Kalies, Vincent Naudot
Horseshoes play a central role in dynamical systems and are observed in many chaotic systems. However most points in a neighborhood of the horseshoe escape after finitely many iterations. In this work we construct a new model by reinjecting the points that escape the horseshoe. We show that this model can be realized within an attractor of a flow arising from a threedimensional vector field, after perturbation of an inclinationflip homoclinic orbit with a resonance. The dynamics of this model, without considering the reinjection, often contains a cuspidal horseshoe with positive entropy, and we show that for a computational example the dynamics with reinjection can have more complexity than the cuspidal horseshoe alone.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.06.009
Issue No: Vol. 334 (2016)
 Authors: Marcus Fontaine; William Kalies; Vincent Naudot
 Rigorous numerics for NLS: Bound states, spectra, and controllability
 Authors: Roberto Castelli; Holger Teismann
Pages: 158  173
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Roberto Castelli, Holger Teismann
In this paper it is demonstrated how rigorous numerics may be applied to the onedimensional nonlinear Schrödinger equation (NLS); specifically, to determining boundstate solutions and establishing certain spectral properties of the linearization. Since the results are rigorous, they can be used to complete a recent analytical proof (Beauchard et al., 2015) of the local exact controllability of NLS.
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.01.005
Issue No: Vol. 334 (2016)
 Authors: Roberto Castelli; Holger Teismann
 Automatic differentiation for Fourier series and the radii polynomial
approach Authors: JeanPhilippe Lessard; J.D. Mireles James; Julian Ransford
Pages: 174  186
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): JeanPhilippe Lessard, J.D. Mireles James, Julian Ransford
In this work we develop a computerassisted technique for proving existence of periodic solutions of nonlinear differential equations with nonpolynomial nonlinearities. We exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system. We compute a numerical Fourier expansion of the periodic orbit for the augmented system, and prove the existence of a true solution nearby using an aposteriori validation scheme (the radii polynomial approach). The problems considered here are given in terms of locally analytic vector fields (i.e. the field is analytic in a neighborhood of the periodic orbit) hence the computerassisted proofs are formulated in a Banach space of sequences satisfying a geometric decay condition. In order to illustrate the use and utility of these ideas we implement a number of computerassisted existence proofs for periodic orbits of the Planar Circular Restricted ThreeBody Problem (PCRTBP).
PubDate: 20160913T04:50:37Z
DOI: 10.1016/j.physd.2016.02.007
Issue No: Vol. 334 (2016)
 Authors: JeanPhilippe Lessard; J.D. Mireles James; Julian Ransford
 Dispersive hydrodynamics: Preface
 Authors: G. Biondini; G.A. El; M.A. Hoefer; P.D. Miller
Pages: 1  5
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): G. Biondini, G.A. El, M.A. Hoefer, P.D. Miller
This Special Issue on Dispersive Hydrodynamics is dedicated to the memory and work of G.B. Whitham who was one of the pioneers in this field of physical applied mathematics. Some of the papers appearing here are related to work reported on at the workshop “Dispersive Hydrodynamics: The Mathematics of Dispersive Shock Waves and Applications” held in May 2015 at the Banff International Research Station. This Preface provides a broad overview of the field and summaries of the various contributions to the Special Issue, placing them in a unified context.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.07.002
 Authors: G. Biondini; G.A. El; M.A. Hoefer; P.D. Miller
 Modulation theory, dispersive shock waves and Gerald Beresford Whitham
 Authors: A.A. Minzoni; Noel F. Smyth
Pages: 6  10
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): A.A. Minzoni, Noel F. Smyth
Gerald Beresford (GB) Whitham, FRS, (13th December, 1927–26th January, 2014) was one of the leading applied mathematicians of the twentieth century whose work over forty years had a profound, formative impact on research on wave motion across a broad range of areas. Many of the ideas and techniques he developed have now become the standard tools used to analyse and understand wave motion, as the papers of this special issue of Physica D testify. Many of the techniques pioneered by GB Whitham have spread beyond wave propagation into other applied mathematics areas, such as reaction–diffusion, and even into theoretical physics and pure mathematics, in which Whitham modulation theory is an active area of research. GB Whitham’s classic textbook Linear and Nonlinear Waves, published in 1974, is still the standard reference for the applied mathematics of wave motion. In honour of his scientific achievements, GB Whitham was elected a Fellow of the American Academy of Arts and Sciences in 1959 and a Fellow of the Royal Society in 1965. He was awarded the Norbert Wiener Prize for Applied Mathematics in 1980.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2015.10.017
 Authors: A.A. Minzoni; Noel F. Smyth
 Dispersive shock waves and modulation theory
 Authors: G.A. El; M.A. Hoefer
Pages: 11  65
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): G.A. El, M.A. Hoefer
There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G.B. Whitham’s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham’s averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for unidirectional (Korteweg–de Vries equation) and bidirectional (Nonlinear Schrödinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including nonclassical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and twodimensional, oblique DSWs.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.04.006
 Authors: G.A. El; M.A. Hoefer
 On the generation of dispersive shock waves
 Authors: Peter D. Miller
Pages: 66  83
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Peter D. Miller
We review various methods for the analysis of initialvalue problems for integrable dispersive equations in the weakdispersion or semiclassical regime. Some methods are sufficiently powerful to rigorously explain the generation of modulated wavetrains, socalled dispersive shock waves, as the result of shock formation in a limiting dispersionless system. They also provide a detailed description of the solution near caustic curves that delimit dispersive shock waves, revealing fascinating universal wave patterns.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.04.011
 Authors: Peter D. Miller
 Dispersive shock waves in the Kadomtsev–Petviashvili and two dimensional
Benjamin–Ono equations Authors: Mark J. Ablowitz; Ali Demirci; YiPing Ma
Pages: 84  98
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Mark J. Ablowitz, Ali Demirci, YiPing Ma
Dispersive shock waves (DSWs) in the Kadomtsev–Petviashvili (KP) equation and two dimensional Benjamin–Ono (2DBO) equation are considered using step like initial data along a parabolic front. Employing a parabolic similarity reduction exactly reduces the study of such DSWs in two space one time ( 2 + 1 ) dimensions to finding DSW solutions of ( 1 + 1 ) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to the cylindrical Korteweg–de Vries (cKdV) and cylindrical Benjamin–Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with very good agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with good agreement. It is concluded that the ( 2 + 1 ) DSW behavior along self similar parabolic fronts can be effectively described by the DSW solutions of the reduced ( 1 + 1 ) dimensional equations.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.01.013
 Authors: Mark J. Ablowitz; Ali Demirci; YiPing Ma
 Whitham theory for perturbed Korteweg–de Vries equation
 Authors: A.M. Kamchatnov
Pages: 99  106
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): A.M. Kamchatnov
Original Whitham’s method of derivation of modulation equations is applied to systems whose dynamics is described by a perturbed Korteweg–de Vries equation. Two situations are distinguished: (i) the perturbation leads to appearance of righthand sides in the modulation equations so that they become nonuniform; (ii) the perturbation leads to modification of the matrix of Whitham velocities. General form of Whitham modulation equations is obtained in both cases. The essential difference between them is illustrated by an example of socalled ‘generalized Korteweg–de Vries equation’. Method of finding steadystate solutions of perturbed Whitham equations in the case of dissipative perturbations is considered.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2015.11.010
 Authors: A.M. Kamchatnov
 Whitham modulation equations, coalescing characteristics, and dispersive
Boussinesq dynamics Authors: Daniel J. Ratliff; Thomas J. Bridges
Pages: 107  116
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Daniel J. Ratliff, Thomas J. Bridges
Whitham modulation theory with degeneracy in wave action is considered. The case where all components of the wave action conservation law, when evaluated on a family of periodic travelling waves, have vanishing derivative with respect to wavenumber is considered. It is shown that Whitham modulation equations morph, on a slower time scale, into the two way Boussinesq equation. Both the 1 + 1 and 2 + 1 cases are considered. The resulting Boussinesq equation arises in a universal form, in that the coefficients are determined from the abstract properties of the Lagrangian and do not depend on particular equations. One curious byproduct of the analysis is that the theory can be used to confirm that the twoway Boussinesq equation is not a valid model in shallow water hydrodynamics. Modulation of nonlinear travelling waves of the complex Klein–Gordon equation is used to illustrate the theory.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.01.003
 Authors: Daniel J. Ratliff; Thomas J. Bridges
 Inverse scattering transform for the defocusing nonlinear Schrödinger
equation with fully asymmetric nonzero boundary conditions Authors: Gino Biondini; Emily Fagerstrom; Barbara Prinari
Pages: 117  136
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Gino Biondini, Emily Fagerstrom, Barbara Prinari
We formulate the inverse scattering transform (IST) for the defocusing nonlinear Schrödinger (NLS) equation with fully asymmetric nonzero boundary conditions (i.e., when the limiting values of the solution at space infinities have different nonzero moduli). The theory is formulated without making use of Riemann surfaces, and instead by dealing explicitly with the branched nature of the eigenvalues of the associated scattering problem. For the direct problem, we give explicit singlevalued definitions of the Jost eigenfunctions and scattering coefficients over the whole complex plane, and we characterize their discontinuous behavior across the branch cut arising from the square root behavior of the corresponding eigenvalues. We pose the inverse problem as a Riemann–Hilbert Problem on an open contour, and we reduce the problem to a standard set of linear integral equations. Finally, for comparison purposes, we present the singlesheet, branch cut formulation of the inverse scattering transform for the initial value problem with symmetric (equimodular) nonzero boundary conditions, as well as for the initial value problem with onesided nonzero boundary conditions, and we also briefly describe the formulation of the inverse scattering transform when a different choice is made for the location of the branch cuts.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.04.003
 Authors: Gino Biondini; Emily Fagerstrom; Barbara Prinari
 Small dispersion limit of the Korteweg–de Vries equation with periodic
initial conditions and analytical description of the Zabusky–Kruskal
experiment Authors: Guo Deng; Gino Biondini; Stefano Trillo
Pages: 137  147
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Guo Deng, Gino Biondini, Stefano Trillo
We study the small dispersion limit of the Korteweg–de Vries (KdV) equation with periodic boundary conditions and we apply the results to the Zabusky–Kruskal experiment. In particular, we employ a WKB approximation for the solution of the scattering problem for the KdV equation [i.e., the timeindependent Schrödinger equation] to obtain an asymptotic expression for the trace of the monodromy matrix and thereby of the spectrum of the problem. We then perform a detailed analysis of the structure of said spectrum (i.e., band widths, gap widths and relative band widths) as a function of the dispersion smallness parameter ϵ . We then formulate explicit approximations for the number of solitons and corresponding soliton amplitudes as a function of ϵ . Finally, by performing an appropriate rescaling, we compare our results to those in the famous Zabusky and Kruskal’s paper, showing very good agreement with the numerical results.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.03.003
 Authors: Guo Deng; Gino Biondini; Stefano Trillo
 Primitive potentials and bounded solutions of the KdV equation
 Authors: S. Dyachenko; D. Zakharov; V. Zakharov
Pages: 148  156
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): S. Dyachenko, D. Zakharov, V. Zakharov
We construct a broad class of bounded potentials of the onedimensional Schrödinger operator that have the same spectral structure as periodic finitegap potentials, but that are neither periodic nor quasiperiodic. Such potentials, which we call primitive, are nonuniquely parametrized by a pair of positive Hölder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded nonvanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.04.002
 Authors: S. Dyachenko; D. Zakharov; V. Zakharov
 On critical behaviour in generalized Kadomtsev–Petviashvili
equations Authors: B. Dubrovin; T. Grava; C. Klein
Pages: 157  170
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): B. Dubrovin, T. Grava, C. Klein
An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev–Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlevé I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture. The numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the (subcritical) KP equations, while in the supercritical KP equations a blowup occurs after the formation of the dispersive shock waves.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.01.011
 Authors: B. Dubrovin; T. Grava; C. Klein
 Semiclassical limit of the focusing NLS: Whitham equations and the
Riemann–Hilbert Problem approach Authors: Alexander Tovbis; Gennady A. El
Pages: 171  184
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Alexander Tovbis, Gennady A. El
The main goal of this paper is to put together: a) the Whitham theory applicable to slowly modulated N phase nonlinear wave solutions to the focusing nonlinear Schrödinger (fNLS) equation, and b) the Riemann–Hilbert Problem approach to particular solutions of the fNLS in the semiclassical (small dispersion) limit that develop slowly modulated N phase nonlinear wave in the process of evolution. Both approaches have their own merits and limitations. Understanding of the interrelations between them could prove beneficial for a broad range of problems involving the semiclassical fNLS.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.03.009
 Authors: Alexander Tovbis; Gennady A. El
 The scattering transform for the Benjamin–Ono equation in the
smalldispersion limit Authors: Peter D. Miller; Alfredo N. Wetzel
Pages: 185  199
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Peter D. Miller, Alfredo N. Wetzel
Using exact formulae for the scattering data of the Benjamin–Ono equation valid for general rational potentials recently obtained in Miller and Wetzel [17], we rigorously analyze the scattering data in the smalldispersion limit. In particular, we deduce precise asymptotic formulae for the reflection coefficient, the location of the eigenvalues and their density, and the asymptotic dependence of the phase constant (associated with each eigenvalue) on the eigenvalue itself. Our results give direct confirmation of conjectures in the literature that have been partly justified by means of inverse scattering, and they also provide new details not previously reported in the literature.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2015.07.012
 Authors: Peter D. Miller; Alfredo N. Wetzel
 The propagation of internal undular bores over variable topography
 Authors: R. Grimshaw; C. Yuan
Pages: 200  207
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): R. Grimshaw, C. Yuan
In the coastal ocean, large amplitude, horizontally propagating internal wave trains are commonly observed. These are long nonlinear waves and can be modelled by equations of the Korteweg–de Vries type. Typically they occur in regions of variable bottom topography when the variablecoefficient Korteweg–de Vries equation is an appropriate model. Of special interest is the situation when the coefficient of the quadratic nonlinear term changes sign at a certain critical point. This case has been widely studied for a solitary wave, which is extinguished at the critical point and replaced by a train of solitary waves of the opposite polarity to the incident wave, riding on a pedestal of the original polarity. Here we examine the same situation for an undular bore, represented by a modulated periodic wave train. Numerical simulations and some asymptotic analysis based on Whitham modulation equations show that the leading solitary waves in the undular bore are destroyed and replaced by a developing rarefaction wave supporting emerging solitary waves of the opposite polarity. In contrast the rear of the undular bore emerges with the same shape, but with reduced wave amplitudes, a shorter overall length scale and moves more slowly.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.01.006
 Authors: R. Grimshaw; C. Yuan
 Nonlinear ring waves in a twolayer fluid
 Authors: Karima R. Khusnutdinova; Xizheng Zhang
Pages: 208  221
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Karima R. Khusnutdinova, Xizheng Zhang
Surface and interfacial weaklynonlinear ring waves in a twolayer fluid are modelled numerically, within the framework of the recently derived 2 + 1 dimensional cKdVtype equation. In a case study, we consider concentric waves from a localised initial condition and waves in a 2D version of the dambreak problem, as well as discussing the effect of a piecewiseconstant shear flow. The modelling shows, in particular, the formation of 2D dispersive shock waves and oscillatory wave trains.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.02.013
 Authors: Karima R. Khusnutdinova; Xizheng Zhang
 Nonlinear disintegration of sine wave in the framework of the Gardner
equation Authors: Oxana Kurkina; Ekaterina Rouvinskaya; Tatiana Talipova; Andrey Kurkin; Efim Pelinovsky
Pages: 222  234
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Oxana Kurkina, Ekaterina Rouvinskaya, Tatiana Talipova, Andrey Kurkin, Efim Pelinovsky
Internal tidal wave entering shallow waters transforms into an undular bore and this process can be described in the framework of the Gardner equation (extended version of the Korteweg–de Vries equation with both quadratic and cubic nonlinear terms). Our numerical computations demonstrate the features of undular bore developing for different signs of the cubic nonlinear term. If cubic nonlinear term is negative, and initial wave amplitude is large enough, two undular bores are generated from the two breaking points formed on both crest slopes (within dispersionless Gardner equation). Undular bore consists of one tabletop soliton and a group of small solitonlike waves passing through the tabletop soliton. If the cubic nonlinear term is positive and again the wave amplitude is large enough, the breaking points appear on crest and trough generating groups of positive and negative solitonlike pulses. This is the main difference with respect to the classic Korteweg–de Vries equation, where the breaking point is single. It is shown also that nonlinear interaction of waves happens similarly to one of scenarios of twosoliton interaction of “exchange” or “overtake” types with a phase shift. If smallamplitude pulses interact with largeamplitude solitonlike pulses, their speed in average is negative in the case when “free” velocity is positive. Nonlinear interaction leads to the generation of higher harmonics and spectrum width increases with amplitude increase independently of the sign of cubic nonlinear term. The breaking asymptotic k 4 / 3 predicted within the dispersionless Gardner equation emerges during the process of undular bore development. The formation of solitonlike perturbations leads to appearance of several spectral peaks which are downshifting with time.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2015.12.007
 Authors: Oxana Kurkina; Ekaterina Rouvinskaya; Tatiana Talipova; Andrey Kurkin; Efim Pelinovsky
 Selffocusing dynamics of patches of ripples
 Authors: P.A. Milewski; Z. Wang
Pages: 235  242
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): P.A. Milewski, Z. Wang
The dynamics of focussing of extended patches of nonlinear capillary–gravity waves within the primitive fluid dynamic equations is presented. It is found that, when the envelope has certain properties, the patch focusses initially in accordance to predictions from nonlinear Schrödinger equation, and focussing can concentrate energy to the vicinity of a point or a curve on the fluid surface. After initial focussing, other effects dominate and the patch breaks up into a complex set of localised structures–lumps and breathers–plus dispersive radiation. We perform simulations both in the inviscid regime and for small viscosities. Lastly we discuss throughout the similarities and differences between the dynamics of ripple patches and selffocussing light beams.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.03.010
 Authors: P.A. Milewski; Z. Wang
 Mechanical balance laws for fully nonlinear and weakly dispersive water
waves Authors: Henrik Kalisch; Zahra Khorsand; Dimitrios Mitsotakis
Pages: 243  253
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Henrik Kalisch, Zahra Khorsand, Dimitrios Mitsotakis
The Serre–Green–Naghdi system is a coupled, fully nonlinear system of dispersive evolution equations which approximates the full water wave problem. The system is known to describe accurately the wave motion at the surface of an incompressible inviscid fluid in the case when the fluid flow is irrotational and twodimensional. The system is an extension of the well known shallowwater system to the situation where the waves are long, but not so long that dispersive effects can be neglected. In the current work, the focus is on deriving mass, momentum and energy densities and fluxes associated with the Serre–Green–Naghdi system. These quantities arise from imposing balance equations of the same asymptotic order as the evolution equations. In the case of an even bed, the conservation equations are satisfied exactly by the solutions of the Serre–Green–Naghdi system. The case of variable bathymetry is more complicated, with mass and momentum conservation satisfied exactly, and energy conservation satisfied only in a global sense. In all cases, the quantities found here reduce correctly to the corresponding counterparts in both the Boussinesq and the shallowwater scaling. One consequence of the present analysis is that the energy loss appearing in the shallowwater theory of undular bores is fully compensated by the emergence of oscillations behind the bore front. The situation is analyzed numerically by approximating solutions of the Serre–Green–Naghdi equations using a finiteelement discretization coupled with an adaptive Runge–Kutta time integration scheme, and it is found that the energy is indeed conserved nearly to machine precision. As a second application, the shoaling of solitary waves on a plane beach is analyzed. It appears that the Serre–Green–Naghdi equations are capable of predicting both the shape of the free surface and the evolution of kinetic and potential energy with good accuracy in the early stages of shoaling.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.03.001
 Authors: Henrik Kalisch; Zahra Khorsand; Dimitrios Mitsotakis
 Traveling waves for a model of gravitydriven film flows in cylindrical
domains Authors: Roberto Camassa; Jeremy L. Marzuola; H. Reed Ogrosky; Nathan Vaughn
Pages: 254  265
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Roberto Camassa, Jeremy L. Marzuola, H. Reed Ogrosky, Nathan Vaughn
Traveling wave solutions are studied for a recentlyderived model of a falling viscous film on the interior of a vertical rigid tube. By identifying a Hopf bifurcation and using numerical continuation software, families of nontrivial traveling wave solutions may be traced out in parameter space. These families all contain a single solution at a ‘turnaround point’ with larger film thickness than all others in the family. In an earlier paper, it was conjectured that this turnaround point may represent a critical thickness separating two distinct flow regimes observed in physical experiments as well as two distinct types of behavior in transient solutions to the model. Here, these hypotheses are verified over a range of parameter values using a combination of numerical and analytical techniques. The linear stability of these solutions is also discussed; both large and smallamplitude solutions are shown to be unstable, though the instability mechanisms are different for each wave type. Specifically, for smallamplitude waves, the region of relatively flat film away from the localized wave crest is subject to the same instability that makes the trivial flatfilm solution unstable; for largeamplitude waves, this mechanism is present but dwarfed by a much stronger tendency to relax to a regime close to that followed by smallamplitude waves.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2015.12.003
 Authors: Roberto Camassa; Jeremy L. Marzuola; H. Reed Ogrosky; Nathan Vaughn
 Interaction of solitons with long waves in a rotating fluid
 Authors: L.A. Ostrovsky; Y.A. Stepanyants
Pages: 266  275
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): L.A. Ostrovsky, Y.A. Stepanyants
Interaction of a soliton with long background waves is studied within the framework of rotation modified Korteweg–de Vries (rKdV) equation. Using the asymptotic method for solitons propagating in the field of a long background wave we derive a set of ODEs describing soliton amplitude and phase with respect to the background wave. The shape of the background wave may range from a sinusoid to the limiting profile representing a periodic sequence of parabolic arcs. We analyse energy exchange between a soliton and the long wave taking radiation losses into account. It is shown that the losses can be compensated by energy pumping from the long wave and, as the result, a stationary soliton can exist, unlike the case when there is no variable background. A more complex case when a free long wave attenuates due to the energy consumption by a soliton is also considered. Some of the analytical results are compared with the results of direct numerical calculations within the framework of the rKdV equation.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.02.008
 Authors: L.A. Ostrovsky; Y.A. Stepanyants
 Observation of dispersive shock waves developing from initial depressions
in shallow water Authors: S. Trillo; M. Klein; G.F. Clauss; M. Onorato
Pages: 276  284
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): S. Trillo, M. Klein, G.F. Clauss, M. Onorato
We investigate surface gravity waves in a shallow water tank, in the limit of long wavelengths. We report the observation of nonstationary dispersive shock waves rapidly expanding over a 90 m flume. They are excited by means of a wave maker that allows us to launch a controlled smooth (single well) depression with respect to the unperturbed surface of the still water, a case that contains no solitons. The dynamics of the shock waves are observed at different levels of nonlinearity equivalent to a different relative smallness of the dispersive effect. The observed undulatory behavior is found to be in good agreement with the dynamics described in terms of a Korteweg–de Vries equation with evolution in space, though in the most nonlinear cases the description turns out to be improved over the quasi linear trailing edge of the shock by modeling the evolution in terms of the integrodifferential (nonlocal) Whitham equation.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.01.007
 Authors: S. Trillo; M. Klein; G.F. Clauss; M. Onorato
 Sine–Gordon modulation solutions: Application to macroscopic
nonlubricant friction Authors: Naum I. Gershenzon; Gust Bambakidis; Thomas E. Skinner
Pages: 285  292
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Naum I. Gershenzon, Gust Bambakidis, Thomas E. Skinner
The Frenkel–Kontorova (FK) model and its continuum approximation, the sine–Gordon (SG) equation, are widely used to model a variety of important nonlinear physical systems. Many practical applications require the wavetrain solution, which includes many solitons. In such cases, an important and relevant extension of these models applies Whitham’s averaging procedure to the SG equation. The resulting SG modulation equations describe the behavior of important measurable system parameters that are the average of the smallscale solutions given by the SG equation. A fundamental problem of modern physics that is the topic of this paper is the description of the transitional process from a static to a dynamic frictional regime. We have shown that the SG modulation equations are a suitable apparatus for describing this transition. The model provides relations between kinematic (rupture and slip velocities) and dynamic (shear and normal stresses) parameters of the transition process. A particular advantage of the model is its ability to describe frictional processes over a wide range of rupture and slip velocities covering seismic events ranging from regular earthquakes, with rupture velocities on the order of a few km/s, to slow slip events, with rupture velocities on the order of a few km/day.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.01.004
 Authors: Naum I. Gershenzon; Gust Bambakidis; Thomas E. Skinner
 Integrable extended van der Waals model
 Authors: Francesco Giglio; Giulio Landolfi; Antonio Moro
Pages: 293  300
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Francesco Giglio, Giulio Landolfi, Antonio Moro
Inspired by the recent developments in the study of the thermodynamics of van der Waals fluids via the theory of nonlinear conservation laws and the description of phase transitions in terms of classical (dissipative) shock waves, we propose a novel approach to the construction of multiparameter generalisations of the van der Waals model. The theory of integrable nonlinear conservation laws still represents the inspiring framework. Starting from a macroscopic approach, a four parameter family of integrable extended van der Waals models is indeed constructed in such a way that the equation of state is a solution to an integrable nonlinear conservation law linearisable by a Cole–Hopf transformation. This family is further specified by the request that, in regime of high temperature, far from the critical region, the extended model reproduces asymptotically the standard van der Waals equation of state. We provide a detailed comparison of our extended model with two notable empirical models such as Peng–Robinson and Soave’s modification of the Redlich–Kwong equations of state. We show that our extended van der Waals equation of state is compatible with both empirical models for a suitable choice of the free parameters and can be viewed as a master interpolating equation. The present approach also suggests that further generalisations can be obtained by including the class of dispersive and viscousdispersive nonlinear conservation laws and could lead to a new type of thermodynamic phase transitions associated to nonclassical and dispersive shock waves.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.02.010
 Authors: Francesco Giglio; Giulio Landolfi; Antonio Moro
 Dispersive shock waves in nematic liquid crystals
 Authors: Noel F. Smyth
Pages: 301  309
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): Noel F. Smyth
The propagation of coherent light with an initial step intensity profile in a nematic liquid crystal is studied using modulation theory. The propagation of light in a nematic liquid crystal is governed by a coupled system consisting of a nonlinear Schrödinger equation for the light beam and an elliptic equation for the medium response. In general, the intensity step breaks up into a dispersive shock wave, or undular bore, and an expansion fan. In the experimental parameter regime for which the nematic response is highly nonlocal, this nematic bore is found to differ substantially from the standard defocusing nonlinear Schrödinger equation structure due to the effect of the nonlocality of the nematic medium. It is found that the undular bore is of Korteweg–de Vries equationtype, consisting of bright waves, rather than of nonlinear Schrödinger equationtype, consisting of dark waves. In addition, ahead of this Korteweg–de Vries bore there can be a uniform wavetrain with a short front which brings the solution down to the initial level ahead. It is found that this uniform wavetrain does not exist if the initial jump is below a critical value. Analytical solutions for the various parts of the nematic bore are found, with emphasis on the role of the nonlocality of the nematic medium in shaping this structure. Excellent agreement between full numerical solutions of the governing nematicon equations and these analytical solutions is found.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2015.08.006
 Authors: Noel F. Smyth
 Incoherent shock waves in longrange optical turbulence
 Authors: G. Xu; J. Garnier; D. Faccio; S. Trillo; A. Picozzi
Pages: 310  322
Abstract: Publication date: 15 October 2016
Source:Physica D: Nonlinear Phenomena, Volume 333
Author(s): G. Xu, J. Garnier, D. Faccio, S. Trillo, A. Picozzi
Considering the nonlinear Schrödinger (NLS) equation as a representative model, we report a unified presentation of different forms of incoherent shock waves that emerge in the longrange interaction regime of a turbulent optical wave system. These incoherent singularities can develop either in the temporal domain through a highly noninstantaneous nonlinear response, or in the spatial domain through a highly nonlocal nonlinearity. In the temporal domain, genuine dispersive shock waves (DSW) develop in the spectral dynamics of the random waves, despite the fact that the causality condition inherent to the response function breaks the Hamiltonian structure of the NLS equation. Such spectral incoherent DSWs are described in detail by a family of singular integrodifferential kinetic equations, e.g. Benjamin–Ono equation, which are derived from a nonequilibrium kinetic formulation based on the weak Langmuir turbulence equation. In the spatial domain, the system is shown to exhibit a large scale global collective behavior, so that it is the fluctuating field as a whole that develops a singularity, which is inherently an incoherent object made of random waves. Despite the Hamiltonian structure of the NLS equation, the regularization of such a collective incoherent shock does not require the formation of a DSW — the regularization is shown to occur by means of a different process of coherence degradation at the shock point. We show that the collective incoherent shock is responsible for an original mechanism of spontaneous nucleation of a phasespace hole in the spectrogram dynamics. The robustness of such a phasespace hole is interpreted in the light of incoherent dark soliton states, whose different exact solutions are derived in the framework of the longrange Vlasov formalism.
PubDate: 20160903T04:24:00Z
DOI: 10.1016/j.physd.2016.02.015
 Authors: G. Xu; J. Garnier; D. Faccio; S. Trillo; A. Picozzi
 Microorganism billiards
 Authors: Saverio E. Spagnolie; Colin Wahl; Joseph Lukasik; JeanLuc Thiffeault
Abstract: Publication date: Available online 18 October 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Saverio E. Spagnolie, Colin Wahl, Joseph Lukasik, JeanLuc Thiffeault
Recent experiments and numerical simulations have shown that certain types of microorganisms “reflect” off of a flat surface at a critical angle of departure, independent of the angle of incidence. The nature of the reflection may be active (cell and flagellar contact with the surface) or passive (hydrodynamic) interactions. We explore the billiardlike motion of a body with this empirical reflection law inside a regular polygon and show that the dynamics can settle on a stable periodic orbit or can be chaotic, depending on the swimmer’s departure angle and the domain geometry. The dynamics are often found to be robust to the introduction of weak random fluctuations. The Lyapunov exponent of swimmer trajectories can be positive or negative, can have extremal values, and can have discontinuities depending on the degree of the polygon. A passive sorting device is proposed that traps swimmers of different departure angles into separate bins. We also study the external problem of a microorganism swimming in a patterned environment of square obstacles, where the departure angle dictates the possibility of trapping or diffusive trajectories.
PubDate: 20161023T18:28:43Z
DOI: 10.1016/j.physd.2016.09.010
 Authors: Saverio E. Spagnolie; Colin Wahl; Joseph Lukasik; JeanLuc Thiffeault
 Modulational instability in a PTsymmetric vector nonlinear
Schrödinger system Authors: J.T. Cole; K.G. Makris Z.H. Musslimani D.N. Christodoulides Rotter
Abstract: Publication date: 1 December 2016
Source:Physica D: Nonlinear Phenomena, Volume 336
Author(s): J.T. Cole, K.G. Makris, Z.H. Musslimani, D.N. Christodoulides, S. Rotter
A class of exact multicomponent constant intensity solutions to a vector nonlinear Schrödinger (NLS) system in the presence of an external P T symmetric complex potential is constructed. This type of uniform wave pattern displays a nontrivial phase whose spatial dependence is induced by the lattice structure. In this regard, light can propagate without scattering while retaining its original form despite the presence of inhomogeneous gain and loss. These constantintensity continuous waves are then used to perform a modulational instability analysis in the presence of both nonhermitian media and cubic nonlinearity. A linear stability eigenvalue problem is formulated that governs the dynamical evolution of the periodic perturbation and its spectrum is numerically determined using Fourier–Floquet–Bloch theory. In the selffocusing case, we identify an intensity threshold above which the constantintensity modes are modulationally unstable for any Floquet–Bloch momentum belonging to the first Brillouin zone. The picture in the selfdefocusing case is different. Contrary to the bulk vector case, where instability develops only when the waves are strongly coupled, here an instability occurs in the strong and weak coupling regimes. The linear stability results are supplemented with direct (nonlinear) numerical simulations.
PubDate: 20161016T12:47:09Z
 Authors: J.T. Cole; K.G. Makris Z.H. Musslimani D.N. Christodoulides Rotter
 On loops in the hyperbolic locus of the complex Hénon map and their
monodromies Authors: Zin Arai
Abstract: Publication date: 1 November 2016
Source:Physica D: Nonlinear Phenomena, Volume 334
Author(s): Zin Arai
We prove John Hubbard’s conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex Hénon map. In fact, we show that there exist several nontrivial loops in the locus which generate infinitely many mutually different monodromies. Furthermore, we prove that the dynamics of the real Hénon map is completely determined by the monodromy of the complex Hénon map, providing the parameter of the map is contained in the hyperbolic horseshoe locus.
PubDate: 20160913T04:50:37Z
 Authors: Zin Arai